It is possible to continue the argument of Theorem REF  a little further: from \(ac = -bd\)  and \(a^{2} + b^{2} = c^{2} + d^{2}\) , it follows that \(a = \pm d\)  and \(b = \mp c\) . So matrices attaining the bound of Theorem REF  are intimately related to sums of two squares. The well-known characterisation of Fermat shows that an integer fails to be a sum of two squares if and only if its square-free part is divisible by a prime \(p \equiv 3 \bmod 4\) ; see, for example, [1]}. From Theorem REF  we obtain matrices meeting the bound of Theorem REF .
